8
votes
1answer
92 views
+100

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
5
votes
1answer
55 views

What is known about the representation theory of the symmetric group over $\mathbb{F}_2$

There is a lot of material available about the representation theory of the symmetric group over $\mathbb{C}$ and fields of characteristic $0$. In particular, there is the decomposition of the group ...
2
votes
1answer
22 views

Decompose the permutation representation into irreducible representations.Construct three non-isomorphic irreducible representations from $S_3$

$S_3$ works on $\mathbb{C^3}$ with the permutation representation. I have to decompose this into irreducible representations and construct three non isomorphic irreducible representations from $S_3$ ...
2
votes
0answers
26 views

Jucys-Murphy elements confusion

I am taking a class called "Harmonic Analysis on Finite Groups" and am studying for an exam. We have recently been talking about the representation theory of the symmetric group (over $\mathbb C$). ...
1
vote
1answer
37 views

Given certain set of symmetries of a tensor, how do you associate the corresponding young tableaux

I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to ...
2
votes
1answer
67 views

Standard represention of $S_3$

I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in the ...
0
votes
0answers
46 views

Obtaining representations of the symmetric group

Consider the following permutation representations three elements in $S_3$: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = ...
0
votes
2answers
49 views

Decomposition of a representation of $S_3$ on monomials

Let $n\ge 2$ be an integer. The symmetric group $S_3$ acts on the set $M_n$ of polynomials in $\mathbb{C}[x_1,x_2,x_3]$ whose monomials are of the form $x_1^{a_1}x_2^{a_2}x_3^{a_3}$ with $0\le a_i\le ...
1
vote
0answers
89 views

Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
0
votes
0answers
42 views

Multiplicity free module and centralizer of subalgebra

This is related to a previous question I posted here. Let $A$ be a semisimple algebra and $B\subset A$ be a subalgebra. Define $$C_A(B) = \{a\in A | ab=ba \quad \forall b\in B\}$$ Let $V$ and ...
3
votes
1answer
72 views

Consequence of the branching rule of S_n representations

Let $V_\lambda$ be the irreducible $S_n$-representation (a left $kS_n$-module) over a field $k$ of characteristic $0$ associated to the partition $\lambda\vdash n$. By abuse of notation let $S_a$ and ...
2
votes
0answers
25 views

The character of $\mathbb{C}[S_n]b_\lambda$

Let $b_\lambda = \sum_{g \in Q_\lambda} (-1)^{sgn(g)}g$, where $Q_\lambda$ is the subgroup of elements that permute the numbers in columns, $\lambda$ a Young diagram corresponding to partition ...
3
votes
1answer
53 views

Number of Cocharge Tableaux Summing to Fixed Numbers

First, some background: I will assume the anglophone conventions for Young tableaux in what follows. Given a standard Young tableau $T$ of shape $\lambda$, we can define the cocharge tableau $C(T)$ as ...
3
votes
1answer
151 views

Definition of the Young Symmetrizer

I have a question regarding the equivalence of two definitions of the young symmetrizer. First, some notation: let $\lambda$ be a partition of $n$. Given a $\lambda$-tableaux $T$, we define the row ...
2
votes
3answers
166 views

Constructing the character table of a group

I am aware that, given a group, there is no simple general procedure to construct the character table of the group (over complex numbers). However, for specific groups, we could use helpful additional ...
7
votes
1answer
223 views

Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
2
votes
4answers
145 views

Irreducible representation of dimension $5$ of $S_5$

i am searching for a concrete as possible description of the (there are two but the are obtained from each other by tensoring with the signature representation) irreducible representation of dimension ...
1
vote
1answer
130 views

Intertwiner of symmetric group representations (Basic)

I am preparing for an exam and there is an excercise which I have to solve but I got stuck. The excercise states: Let $V=\mathbb{C}^3$ be the permutation representation of the symmetric group $S_3$. ...
3
votes
0answers
83 views

On Applications of the Murnagham Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnagham Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
4
votes
2answers
86 views

Splitting fields of symmetric groups

Is it true that $k$ is a splitting field of $S_n$ if and only if the characteristic $p$ of $k$ is zero or larger than $n$? The fact that the character table (over $\mathbb C$) has only integer entries ...
3
votes
1answer
45 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
1
vote
1answer
109 views

Are all of the irreducible representations of (any) symmetric group over $\mathbb{C}$ also irreducible over a finite splitting field.

This is probably an incredibly stupid question, but I'm a novice to representation theory and finite field theory, as I've just been introduced to these concepts, so all I really need is confirmation ...
6
votes
0answers
115 views

Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
0
votes
1answer
66 views

Specific question on Sn modules

Let $L_{-1}$ denote the 1-dimensional sign-representation of the symmetric group $S_n$ and V the standard $(n - 1)$-dimensional module for $S_n$. How to prove that V and $V \otimes L_{-1}$ are not ...
1
vote
0answers
61 views

Quote on the Littlewood-Richardson Rule

In Gordon James's paper "The representation Theory of the Symmetric Group" he says "The author was once told that the Littlewood–Richardson rule helped to get men on the moon but was not proved until ...
6
votes
0answers
82 views

Characters of the symmetric group corresponding to partitions into two parts

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
5
votes
1answer
88 views

What is the centralizer of the Young symmetrizer?

I have read a lot about idempotents, several important facts were about central idempotents. Now, the Young symmetrizer is a constant away from an idempotent, but I don't think it's central. ...
6
votes
1answer
166 views

Some irreducible characters of the Symmetric group $S_n$

I want to have characters of some irreducible $S_n$-modules corresponding to certain partitions $\lambda$ of $n$, the computations using Frobenius formula get complicated and I am unable to find in ...
8
votes
2answers
287 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
4
votes
2answers
122 views

Law of large numbers for Plancherel random Young diagrams

Do you know a reference book on the law of large numbers for random Plancherel Young diagrams ? I know the book of Kerov, but actually, it is only a compilation of his articles, and i need something ...
8
votes
1answer
190 views

Schur -Weyl duality for $sl_2$ and $S_n$

$V$ is an $m$ dimensional vector space having a structure of $sl_2(\mathbb{C})$-module, where $sl_2(\mathbb{C})$ is the Lie algebra of the Lie group $SL_2(\mathbb{C})$. The symmetric group $S_n$ acts ...
6
votes
0answers
186 views

Invariant element in the tensor product of rectangular Specht modules?

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
1
vote
2answers
347 views

Convolution of irreducible characters of a finite group

If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $$ \chi^{\lambda} ...
3
votes
0answers
107 views

Character formula for $S_n$ and $GL(V)$

In a set of lecture notes I'm reading, we consider representations of the symmetric group $S_n$ via treatment of Young tableaux, partitions of $n$ etc. (in what I believe is the standard approach) - ...
1
vote
2answers
664 views

Irreducibility of the standard representation of $S_n$.

The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
14
votes
1answer
404 views

Low-dimensional Irreducible Representations of $S_n$

For $n\geq 7$, I would like to show that $S_n$ has no irredicuble representations of dimension $m$ for $2\leq m\leq n-2$. The catch is that I am not allowed to use any "machinery" (evidently, this ...
14
votes
2answers
272 views

A question on partitions of n

Let $P$ be the set of partitions of n. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : ...
2
votes
1answer
227 views

Centre of symmetric group algebra

I'd like to know a reference for a simple proof that $\{c_\mu\mid \mu\vdash n\}$ is a basis for the centre of the symmetric group algebra $\mathbb{C}\mathfrak{S}_n$, where $c_\mu$ is the sum of all ...
3
votes
1answer
279 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
4
votes
2answers
288 views

How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
9
votes
1answer
324 views

Field of definition of representations of symmetric groups

Can one show in an elementary way, without recourse to Young tableaux etc., that the complex representations of symmetric groups are realisable over $\mathbb{R}$? It is easy to show that they are all ...